A note on 2-distant noncrossing partitions and weighted Motzkin paths

We prove a conjecture of Drake and Kim: the number of 2-distant noncrossing partitions of {1,2,…,n} is equal to the sum of weights of Motzkin paths of length n, where the weight of a Motzkin path is a product of certain fractions involving Fibonacci numbers. We provide two proofs of their conjecture: one uses continued fractions and the other is combinatorial.     

Decay of correlations for Fibonacci unimodal interval maps

We consider a class of generalized Fibonacci unimodal maps for which the central return times {s_n} satisfy that s_n = s_(n-1) +ks_(n-2) for some k ≥ 1. We show that such a unimodal map admits a unique absolutely continuous invariant probability with exactly stretched exponential decay of correlations if its critical order lies in(1, k+1).     

关于反共轭有序分拆的几个恒等式

In this paper, we first present several identities related to the inverse-conjugate compositions having parts of size ≤ 3, the compositions into parts equal to 1 or 2, the compositions into odd parts and the compositions into parts greater than 1. In addition, we provide a bijective proof of a relation for inverse-conjugate compositions having parts of size ≤ k.     

二阶微分方程的逆特征值问题

运用渐近分析的方法及Rayleigh商原理,将Sturm-Liouville问题的Ambarzumyan定理推广到具有Neumann边界条件或拟周期边界条件的二阶微分方程情形.同时,获得了二阶向量微分方程的有关Ambarzumyan型结果.     

强symmetric环

为了统一交换环和约化环的层表示,Lambek引进了Symmetric环.继续symmetric环的研究,定义引入了强symmetric环的概念,研究它的一些扩张性质.证明环R是强symmetric环当且仅当R[x]是强symmetric环当且仅当R[x;x~(-1)]是强symmetric环.也证明对于右Ore环R的经典右商环Q,R是强symmetric环当且仅当Q是强symmetric环.     

A note on harmonic Ritz values and their reciprocals

This note summarizes an investigation of harmonic Ritz values to approximate the interior eigenvalues of a real symmetric matrix A while avoiding the explicit use of the inverse A^?1. We consider a bounded functional ψ that yields the reciprocals of the harmonic Ritz values of a symmetric matrix A. The crucial observation is that with an appropriate residual s, many results from Rayleigh quotient and Rayleigh–Ritz theory naturally extend. The same is true for the generalization to matrix pencils (A, B) when B is symmetric positive definite. These observations have an application in the computation of eigenvalues in the interior of the spectrum of a large sparse matrix. The minimum and maximum of ψ correspond to the eigenpairs just to the left and right of zero (or a chosen shift). As a spectral transformation, this distinguishes ψ from the original harmonic approach where an interior eigenvalue remains at the interior of the transformed spectrum. As a consequence, ψ is a very attractive vehicle for a matrix‐free, optimization‐based eigensolver. Instead of computing the smallest/largest eigenvalues by minimizing/maximizing the Rayleigh quotient, one can compute interior eigenvalues as the minimum/maximum of ψ. Copyright © 2009 John Wiley & Sons, Ltd.     

Two-Dimensional Töplitz Operators with Measurable Symbols

For the complete algebra of two-dimensional Töplitz operators with measurable bounded symbols, we establish conditions necessary for the Fredholm property of the operators and prove results on the separation of singularities of the symbols. As a corollary, conditions sufficient for the Fredholm property are established for operators with symbols satisfying local conditions of sectorial type.     

Estimating perceived phonatory pressedness in singing from flow glottograms

The normalized amplitude quotient (NAQ), defined as the ratio between the peak-to-peak amplitude of the flow pulse and the negative peak amplitude of the differentiated flow glottogram and normalized with respect to period time, has been shown to be related to glottal adduction. Glottal adduction, in turn, affects mode of phonation and hence perceived phonatory pressedness. The relationship between NAQ and perceived phonatory pressedness was analyzed in a material collected from a professional female singer and singing teacher who sang a triad pattern in breathy, flow, neutral, and pressed phonation in three different loudness conditions (soft, middle, loud). In addition, she also sang the same triad pattern in four different styles of singing, classical, pop, jazz, and blues, in the same three loudness conditions. A panel of experts rated the degree of perceived phonatory press along visual analogue scales. Comparing the obtained mean rated pressedness ratings with the mean NAQ values for the various triads showed that about 73% of the variation in perceived pressedness could be accounted for by variations of NAQ.     

On the limit-classifications of even and odd-order formally symmetric differential expressions

In this paper we consider the formally symmetric differential expressionM [.] of any order (odd or even) ≥ 2. We characterise the dimension of the quotient spaceD(T _max)/D(T _min) associated withM[.] in terms of the behaviour of the determinants det [[f _rg_s](∞)] where 1 ≤n ≤ (order of the expression +1); here [fg](∞) = lim [fg](x), where [fg](x) is the sesquilinear form in f andg associated withM. These results generalise the well-known theorem thatM is in the limit-point case at ∞ if and only if [fg](∞) = 0 for everyf, g ε the maximal domain Δ associated withM.     

偏代数族的积偏代数

定义了一族偏代数物积偏代数的概念，且对偏代数间的同态映射，由同态映射诱导的合同关系及由同态映射和合同关系诱导的合同关系与积偏代数一起作了深入的讨论，分别得到了一些有意义的结果。